A particle of mass kg moves in a plane under the isotropic potential with

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables

Problem

A particle of mass

\mu = 0.50

kg moves in a plane under the isotropic potential

V(r) = \tfrac12\kappa r^2

with

\kappa = 8.0\,\mathrm{N\,m^{-1}}

. Its energy is

E = 2.0

J and its angular momentum about the centre is

\ell = 0.40\,\mathrm{kg\,m^2\,s^{-1}}

. ((a)) Write the Hamiltonian in plane polar coordinates

(r,\varphi)

, write

S = W(r,\varphi) - Et

, and separate the Hamilton-Jacobi equation, identifying the constants

E

and

\ell

and writing the radial equation for

p_r(r)

. ((b)) For this potential the radial loop integral is a standard result,

I_r = \dfrac{E}{2\omega} - \dfrac{\ell}{2}

with

\omega = \sqrt{\kappa/\mu}

. Find the energy

E_{\rm circ}

of the circular orbit that has this same

\ell

(the orbit with

I_r = 0

), and compare it with the given

E

. ((c)) Write

H

in terms of

I_r

and

I_\varphi = \ell

, and use

\nu_k = \partial H/\partial I_k

to find the radial frequency

\nu_r

and the angular frequency

\nu_\varphi

. ((d)) Hence find the angle swept from one closest approach to the next, and state whether the orbit is closed.

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